Title:
Towards the fast scrambling conjecture

Abstract: Many proposed quantum mechanical models of black holes include highly
nonlocal interactions. The time required for thermalization to occur in such
models should reflect the relaxation times associated with classical black
holes in general relativity. Moreover, the time required for a particularly
strong form of thermalization to occur, sometimes known as scrambling,
determines the time scale on which black holes should start to release
information. It has been conjectured that black holes scramble in a time
logarithmic in their entropy, and that no system in nature can scramble faster.
In this article, we address the conjecture from two directions. First, we
exhibit two examples of systems that do indeed scramble in logarithmic time:
Brownian quantum circuits and the antiferromagnetic Ising model on a sparse
random graph. Unfortunately, both fail to be truly ideal fast scramblers for
reasons we discuss. Second, we use Lieb-Robinson techniques to prove a
logarithmic lower bound on the scrambling time of systems with finite norm
terms in their Hamiltonian. The bound holds in spite of any nonlocal structure
in the Hamiltonian, which might permit every degree of freedom to interact
directly with every other one.